In this paper, we study MBP-preserving and energy dissipative schemes for the Allen–Cahn type equations
by combining the ESAV approach with stabilizing technique.
We present first- and second-order sESAV schemes
and prove their MBP preservation, energy dissipation, and error estimates.
The main results and observations include two aspects.
First, we choose the ESAV approach rather than the classic SAV approach,
since the coefficient (\(g(u^n,s^n)\)  or \( g(\widehat{u}^{n+\frac{1}{2}},\widehat{s}^{n+\frac{1}{2}})\) )
of the nonlinear term is positive automatically in the former one
while the sign of the corresponding coefficient is uncertain for the later one.
Second, to guarantee the MBP-preserving property,
we add the stabilization term as an extra artificial term,
that is, add and subtract a linear term in the scheme instead of a quadratic term in the energy functional;
they are equivalent mutually for the classic stabilization or convex splitting method,
but not for the SAV approach. Moreover, we find that
the MBP preservation and the energy dissipation of the sESAV schemes
can be established in parallel and independently,
unlike the purely stabilized semi-implicit scheme discussed in [1]}
where the MBP is needed first to bound the nonlinear term
in the proof of the stability with respect to the original energy.
Since the schemes we studied are all one-step methods,
adaptive time-stepping strategies (such as [2]})
can be inherently adopted to accelerate the computation.
